# Linear functions and slope on the SAT

Canonical: https://duckyhelper.com/learn/sat-math/linear-functions/
Updated: 2026-10-01

A linear function changes by the same amount every step, so it can be written \(f(x) = mx + b\). The slope \(m\) is the rate of change: rise over run, or the "per" number in a word problem. The intercept \(b\) is the starting value, the output when x is 0. SAT questions ask you to find m and b from two points, a table or a graph, and to say what they mean.

## The key idea

Slope is change in output divided by change in input. Take any two points on the line, subtract in the same order on top and bottom:

$$
m = \frac{y_2 - y_1}{x_2 - x_1} \qquad f(x) = mx + b
$$

In a word problem, the slope always has units of "output per input": dollars per hour, centimeters per week, gallons per minute. The intercept has the output's units: dollars, centimeters, gallons.

**Three ways the SAT writes the same line**

| Form | Looks like | What you can read off |
| --- | --- | --- |
| Slope-intercept | \(y = mx + b\) | Slope \(m\), y-intercept \(b\) |
| Point-slope | \(y - y_1 = m(x - x_1)\) | Slope \(m\), a point \((x_1, y_1)\) |
| Standard | \(Ax + By = C\) | Slope \(-\frac{A}{B}\), y-intercept \(\frac{C}{B}\), x-intercept \(\frac{C}{A}\) |

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals, so their product is \(-1\): a slope of \(\frac{2}{3}\) pairs with \(-\frac{3}{2}\).

## Worked examples

**Example 1: the line through two points**

Problem: Find the equation of the line through \((2, 5)\) and \((6, 17)\).

1. Find the slope. Subtract in the same order on top and bottom.

   $$
   m = \frac{17 - 5}{6 - 2} = \frac{12}{4} = 3
   $$
2. Put one point into \(y = 3x + b\) to find b. Using \((2, 5)\):

   $$
   5 = 3(2) + b
   $$
3. Solve for b.

   $$
   b = -1
   $$
4. Check with the other point: \(3(6) - 1 = 17\). It works.

Answer: \(y = 3x - 1\)

**Example 2: build the function from a table, then explain it**

Problem: A candle burns at a steady rate. After 1 hour it is 21 cm tall, after 3 hours 15 cm, and after 6 hours 6 cm. Write the height \(h(t)\) after t hours and say what each number means.

1. Slope from the first two rows: change in height over change in time.

   $$
   m = \frac{15 - 21}{3 - 1} = -3
   $$
2. Make sure the third row fits the same rate: from hour 3 to hour 6 the height drops 9 cm in 3 hours, which is also \(-3\) per hour.
3. The intercept is the height at \(t = 0\), not at \(t = 1\). Go back one hour from 21 cm by adding 3.

   $$
   h(0) = 21 + 3 = 24
   $$
4. So \(h(t) = -3t + 24\). The candle started 24 cm tall and loses 3 cm each hour. It burns out when \(h(t) = 0\).

   $$
   -3t + 24 = 0
   $$

Answer: \(h(t) = -3t + 24\): it starts at 24 cm, shrinks 3 cm per hour, and burns out after 8 hours.

**Example 3: a perpendicular line from standard form**

Problem: Line k is perpendicular to the line \(4x - 6y = 9\) and passes through \((4, -1)\). What is the y-intercept of line k?

1. Slope of the given line is \(-\frac{A}{B}\), with \(A = 4\) and \(B = -6\).

   $$
   m = -\frac{4}{-6} = \frac{2}{3}
   $$
2. Perpendicular slope: flip it and change the sign.

   $$
   m_k = -\frac{3}{2}
   $$
3. Use \(y = -\frac{3}{2}x + b\) with the point \((4, -1)\).

   $$
   -1 = -\frac{3}{2}(4) + b
   $$
4. So \(-1 = -6 + b\).

   $$
   b = 5
   $$

Answer: The y-intercept is 5, at the point \((0, 5)\).

**Example 4 (SAT-hard): skip the equation**

Problem: For a linear function f, \(f(2) = 11\) and \(f(5) = 20\). What is \(f(10) - f(4)\)?

1. Slope from the two given points.

   $$
   m = \frac{20 - 11}{5 - 2} = 3
   $$
2. For a line, a change in output is always slope times change in input. The inputs 10 and 4 are 6 apart.

   $$
   f(10) - f(4) = 3 \cdot 6 = 18
   $$
3. Check by building f: \(f(x) = 3x + 5\), so \(f(10) = 35\) and \(f(4) = 17\), and \(35 - 17 = 18\).

Answer: 18

## Common mistakes

- **Mixing the subtraction order.** \(\frac{17 - 5}{2 - 6}\) gives the wrong sign. Fix: whichever point goes first on top goes first on the bottom too.
- **Calling A the slope in \(Ax + By = C\).** The slope of \(4x - 6y = 9\) is \(\frac{2}{3}\), not 4. Fix: solve for y, or use \(-\frac{A}{B}\).
- **Using the first table row as the starting value.** If the table starts at \(t = 1\), that row is not the intercept. Fix: the intercept is the output when the input is exactly 0.
- **Forgetting the sign on a perpendicular slope.** The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\), but perpendicular needs \(-\frac{3}{2}\). Fix: check that the two slopes multiply to \(-1\).
- **Explaining the slope with the wrong units.** In \(h(t) = -3t + 24\), 3 is centimeters per hour, not hours per centimeter. Fix: say "output units per one input unit".

## Quick methods

> **Tip: Differences without the equation**
>
> For a line, \(f(a) - f(b) = m(a - b)\). When a question asks how much the output changes, you only need the slope. This is exact, not an estimate.

> **Note: Desmos regression for tables**
>
> In Desmos, add a table, enter the x and y values, then type \(y_1 \sim mx_1 + b\). If the data is exactly linear, Desmos shows the exact m and b. With two clean points, doing it by hand is just as fast. More in [Desmos on the SAT](https://duckyhelper.com/learn/sat-math/desmos-calculator/).

## Practice

**5 SAT-style questions**

1. What is the slope of the line through \((-3, 4)\) and \((5, -8)\)?
   A. \(-\frac{3}{2}\)
   B. \(-\frac{2}{3}\)
   C. \(\frac{3}{2}\)
   D. \(-6\)

   Answer: \(-\frac{3}{2}\). Rise over run: \(\frac{-8 - 4}{5 - (-3)} = \frac{-12}{8} = -\frac{3}{2}\). The run is 8, not 2: subtracting a negative adds. \(-\frac{2}{3}\) puts run over rise.

2. What is the slope of the line \(3x + 5y = 30\)?
   A. \(-\frac{3}{5}\)
   B. \(\frac{3}{5}\)
   C. \(-\frac{5}{3}\)
   D. \(3\)

   Answer: \(-\frac{3}{5}\). Solve for y: \(5y = -3x + 30\), so \(y = -\frac{3}{5}x + 6\). The 3 is the x coefficient in standard form, not the slope.

3. The water in a tank, in gallons, t minutes after a pump starts is \(W(t) = 850 - 25t\). What is the best interpretation of 25?
   A. The tank starts with 25 gallons.
   B. The pump removes 25 gallons each minute.
   C. The tank is empty after 25 minutes.
   D. The pump adds 25 gallons each minute.

   Answer: The pump removes 25 gallons each minute.. 25 is multiplied by t, so it is a rate, and its minus sign means the water goes down. The tank starts with 850 gallons, and it empties after \(850 \div 25 = 34\) minutes.

4. Which line is perpendicular to \(y = 4x - 7\)?
   A. \(y = 4x + 7\)
   B. \(y = -4x + 1\)
   C. \(y = -\frac{1}{4}x + 3\)
   D. \(y = \frac{1}{4}x - 7\)

   Answer: \(y = -\frac{1}{4}x + 3\). Perpendicular slopes multiply to \(-1\): \(4 \cdot (-\frac{1}{4}) = -1\). \(y = 4x + 7\) is parallel. A slope of \(-4\) changes only the sign, and \(\frac{1}{4}\) only flips the number. You need both changes.

5. Student-produced response: a linear function p has \(p(1) = -2\) and \(p(6) = 18\). What is \(p(0)\)?

   Answer: -6. Slope: \(\frac{18 - (-2)}{6 - 1} = 4\). Going from \(x = 1\) back to \(x = 0\) subtracts one slope: \(-2 - 4 = -6\). So \(p(x) = 4x - 6\).

## Frequently asked questions

### Is slope the same as rate of change?

For a line, yes. The slope is the rate of change, and it is the same between any two points. For curves like parabolas the rate keeps changing, which is one way to tell a linear function from a nonlinear one in a table.

### How do I find the slope from standard form?

For \(Ax + By = C\), the slope is \(-\frac{A}{B}\). If you do not trust the shortcut, solve for y: subtract Ax from both sides and divide by B. The number in front of x is the slope.

### What does the y-intercept mean in a word problem?

It is the value of the output when the input is 0: the starting amount, the flat fee, or the height at time zero. Say it with units, like "the plan costs $12 before any texts are sent".

### How do I tell if two lines are parallel or perpendicular?

Write both in \(y = mx + b\) form and compare slopes. Equal slopes with different intercepts mean parallel. Slopes that multiply to \(-1\) mean perpendicular. Equal slopes and equal intercepts mean it is the same line.

## Sources

- [College Board: SAT Math, Algebra skills](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/algebra), accessed 2026-10-01

## Related

- [Linear equations in one variable on the SAT](https://duckyhelper.com/learn/sat-math/linear-equations/)
- [Systems of linear equations on the SAT](https://duckyhelper.com/learn/sat-math/systems-of-equations/)
- [Slope-intercept form and how to write the equation of a line](https://duckyhelper.com/learn/algebra-1/slope-intercept-form/)
- [How to find the slope of a line](https://duckyhelper.com/learn/algebra-1/slope/)
- [SAT Math study guides](https://duckyhelper.com/learn/sat-math/)

## Try asking Ducky

- "Why is the starting height 24 and not 21?"
- "What does the slope mean in this problem, in plain words?"
- "Give me a table and let me find the equation myself."
- "I keep getting the sign of the slope wrong. Can you watch me do one?"

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